## N

For x ^ 0, Equations (6.17) and (6.24) coincide and are independent of the surface fractal dimension. However, in contrast to Equation (6.17), the term x2 resulting from Equation (6.24) is independent of the fractal dimension. The region x < 0.15 is of interest for the determination of the monolayer capacity. Thus, for any value of Ds, monolayer capacities Nm can be determined from the standard BET method as long as the experimental measurements are restricted to the range of low relative pressures.

Both Equations (6.22) and (6.25) predict that, for x ^ 1, the growth of the film is described by the power law ln(-N) a (D - 3) ln(1 - x) (6.28)

Film growth is independent of the substrate-film interaction, i.e. the parameter C. There is a simple heuristic justification of the exponent Ds - 3 in the power law given above [3]. On a planar surface, the mean film thickness for x ^ 1 can be estimated as ct/(1 -x). This suggests that, for an arbitrary surface, the film volume is V [ct/(1 -x)], where V(z) is the volume of all points located within the film at distances lower than or equal to z. Since the derivative dV(z)/dz can be interpreted as the surface area measured by molecules of diameter z, d V(z)/dz |z=o should be equal to the specific surface area Nmo2. Therefore, from Equation (6.17), it is possible to deduce that for a fractal surface and for z o:

where B is a constant. After substitution of z = o/(1 - x), the power law is recovered in Equation (6.25) from Equation (6.29). For z values down to o, Equation (6.29) can be corrected by including an additional term that ensures correct behavior at z= o:

The constants B and B' can be evaluated when the value of the derivative of the function V(z) at z = o is V'(o) = Nmo2 and when V(o) = Nmo3 is the volume of the monolayer. These two conditions uniquely fix the values of B and B'. Consequently:

Nm o3

The dependence of the Equations (6.17) and (6.24) and Equations (6.22) and (6.25) on the surface fractal dimension Ds offers a method for measuring Ds from a single experimental isotherm. This may be done by fitting Equations (6.22) or (6.25) to the data obtained at relative pressures close to unity. In such a case, it is necessary to determine how large x should be in order for the asymptotic Equations (6.17) and (6.25) to be valid. A simple estimate proposed by Pfeifer et al. [3] follows from the requirements that N/Nm > 2, since the adsorption energy of molecules adsorbed in the second and higher layers is independent of the energy of interaction with the solid substrate. This estimate yields x > 1 - 2-1/(3-Ds) as a minimal condition for the relative pressure x to be in the asymptotic regime. For Ds = 2.5, this condition gives x > 0.75, whereas the asymptotic regime is shifted towards higher relative pressures for larger values of Ds.

To make a definitive statement about the fractal dimension and the length range, it would be useful to carry out analyses of adsorption data evaluated using different adsorbates. Unfortunately, such analyses are not always possible. Furthermore, since different adsorbate molecules can only be specifically adsorbed by a given type of site, some parts of the adsorbent are left almost completely uncovered and the values of Ds obtained for different adsorbates may be different.

Equations (6.13) and (6.16) were developed to describe adsorption on a surface of topological dimension Dtop = 2, in three-dimensional space d = 3. Equation (6.17) can also be generalized to the case of an arbitrary topological and space dimension [3]. This generalization may be important because, in the case of the adsorption on fractal soil materials including clusters of aggregated colloids, polymeric substances (e.g. humic substances), etc., Dtop = 1 rather than Dtop = 2. Finally, it should be noted that fractal analogues of the BET isotherm have been developed by some authors [49, 51-54].

### 6.3.2 Frenke-Halsey-Hill Theory

The Frenkel-Halsey-Hill (FHH) isotherm was originally developed to describe the growth of thick films and wetting phenomena on a flat surface and was later extended to studying adsorption on fractal surfaces [3, 55]. In contrast to BET theory, FHH theory applies to long-range adsorbate-absorbent interactions and its approach is closely related to the so-called potential theory of adsorption of Eucken and Polanyi (see Ref. [35]).

A common method for deriving the FHH adsorption isotherm is to determine the chemical potential of an adsorbate at a distance z from the substrate with respect to that in the bulk state. If the adsorbing surface is flat and if the adsorbed film is structurally similar to the bulk liquid adsorbate at a given temperature, then the difference between the chemical potentials, A is

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